Turing instability and Hopf bifurcation for a diffusion-plankton system with cell size

Abstract

This paper investigates Turing instability and Hopf bifurcation for a diffusive plankton system with time delay and cell size. To determine the effects of diffusion and cell size on the dynamics of the system, we first study the system without time delay, where the conditions of stability of coexisting equilibrium and Turing instability are obtained through Routh-Hurwitz criterion. Then we give the existence of Hopf bifurcation using time delay as bifurcation parameter by analyzing the distribution of eigenvalues, and derive the property of Hopf bifurcation by applying the centre manifold theory. Finally, numerical simulation shows that different cell size increases the variety of dynamics for diffusive plankton system.

Keywords:

• Diffusive plankton system
• Turing instability
• Hopf bifurcation
• cell size
• time delay

2010 Mathematics Subject Classifications:

• 35K57
• 37N25
• 92D25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by National Natural Science Foundation of China (Nos. U1806203, 61533011, 61903343).